(circa 2018) These notes are just some layman’s “armchair pseudo-philosophy”, however they have been quite successful in calming down all my previous “metaphysical anxieties” (especially concerning mathematical objects and ways of experiencing them). Also they make me happy and fed, and seem to organize quite well all of my experiences which I happen to remember — including dreams, hallucinations, fantasies, abstract thoughts etc.
I am sitting in a room and thinking about Well Ordering Principle: that any set, disregarding its cardinality, can be well-ordered. That would be splendid as then any choice procedure would be possible, so I could e.g. extend any filter to ultrafilter. I have no doubt that well ordering assures me these goods, though given the basic facts about forming sets I can not figure out if the well ordering is always possible or not.
(N.b. the “basic facts” are just that a pair of any two sets can always be formed, same for union of any family of sets, powerset of any set, any set’s projection via any [first order] predicate, and that there is at least one infinite set)
It is rather obvious I am thinking about abstract objects: there are no such things as sets, filters or well-orderings, and even if there were, no physical space that I can perceive in finite time would fit the transfinite set I’d like to well-order, and no finite amount of time would allow me to do it by any thinkable or unthinkable activity… Infinite sets are a fairy-tale, just as numbers, names, relations, functions, points and planes, and structures and their morphisms, computer programs and algorithms. As well as forces, masses, accelerations, trajectories. Though completely unreal, these abstractions are very useful: they allow me to make (sometimes even accurate) predictions about finite objects: behaviour of a computer running some particular program, the tension inside a piece of construction, a trajectory of a cannonball etc.
Writing these words I stop to grab my mug with coffee to take a sip. I have no problem with its identity over the time: until it breaks it remains — no doubt — my mug, and it contains coffee, the very same coffee I poured in some time ago, only colder (somehow it is the same coffee, but with different temperature).
However assuming naive physicalism I know there is no such thing as a mug: there is rather some denser cloud of tiny vibrating particles in another, sparser cloud, exchanging particles with the surrounding etc. It is a series of different, perhaps even unrelated phenomena; there is no mug nor coffee, or cannonball or computer — they are some nice abstractions, quite useful for keeping track of my coffee consumption, software development, ballistics… of organizing my experience in general.
But of course there is also no such things as particles, are there? The concept of an atom is a nice extrapolation of various experiences with balls, balloons, perhaps some metaphor of a pool table too. It does not really help these supposed particles are orbited by some even tinier balls with negative charge (and even worse: not exactly located anywhere)… And what is this charge anyway? This is all just a nice, helpful function, an elaborate fairy tale loosely connected to my experiences, a metaphor.
So it turns out my entire experience I can organize into various “levels”: from considerations of micro-physics, through “everyday objects”, computer programs, up to the level of infinite sets. All are actually fictional and abstract, and all of these notions are partly predictable to me:
“there is no coffee left in the mug”,
“this program always terminates”,
“there are two electrons on the orbit of helium particle”,
“there is no bijection between any set and its powerset”,
…and partly a mystery, e.g.:
“what, I drank all the coffee?”,
“it turns out there is a bug in my software!”,
“where is the electron now?”,
“can there be a set so big it can’t be well-ordered?”
Do these levels form any particular hierarchy? I do not believe so. Despite how I framed it above, there is no reason to assume programs are below sets and mugs are below programs; or above, or in any particular, even non-linear order… I can fairly well rearrange this hierarchy and each one seems good enough: I can predict some facts about sets thinking about mugs, and predict some properties of mugs thinking about sets for example. This goes even further: I can take any particular “level” and reject all the other ones and still function quite well, at least for some time (which is what I actually do as my mind is too small to think on all those levels simultaneously).
Do these levels have anything in common though? Surely there is myself, the one who thinks and utters them. But it is my guess that there is also some structural, architectural common denominator to them all: namely, I think about them all (or rather: “I think them all”!) using the same “grammar”: a certain framework which splits or groups or structures the stories into statements, and statements into objects and relations — noun phrases and verb phrases, decorated with various attributions and complementations… They are literally stories: utterances of my inner dialogue.
Extending Korzybski’s metaphor “the map is not the territory” I necessarily come to the conclusion there is no territory at all! There are only maps of maps of maps, maps all the way down, all painted with the same grammatical structure. At any moment I may choose one such map and treat it as the territory, but that “territory” is nothing more than some imagination of what the map in front of me might represent. And there is usually no problem to swap the maps, e.g. to explain a ball with particles, a particle with ball, sets with mugs or mugs with sets.
To put it all in simple worlds: it seems I live in stories, grammatical structures, which paint all of my experience and allow any perception and reasoning.
This leads to three rather interesting questions:
Is all of my thinking purely grammatical, or is there some ``geometric thinking’’ which would be neither grammatical, nor expressible grammatically?
What do these “worlds” or stories relate to? Are they merely self-contained systems?
are those stories, “local realities”, always consistent, and can they be incomplete?
Ad 1. I believe there are no non-grammatical thoughts. That belief is loosely based on my self-reflections (perhaps others have it differently but given the above “others” are not, since “others” is just a noun phrase), and the fact that Euclidean geometry is a decidable first order theory. If that is not a good reason, at least it sounds pretty seriously. And of course there are other hints in this direction: certain perfectly well stories are “not geometric”, i.e. not relatable to kinestetic senses as a whole image. One neat example is a story by Ronald Topor about a dishonest dentist, which went roughly like this:
A man visits a dentist, dentist examines his teeth, finds a crack, puts a finger in it, then entire hand, eventually tells the man to keep his mouth open and walks into the teeth. The man hears fading echo of dentist’s footsteps. After about an hour the dentist is back and says “well I’m afraid we have to remove the entire teeth”; he removes the teeth and charges the man for the procedure. Only later it turned out the dentist discovered well preserved ancient frescos inside the teeth, which he kept it in his clinic and was allowing visitors in — for money of course.
It makes a perfectly understandable story.
On the other hand, it would be unfair if I did not take into consideration that if there is some non-grammatical thinking, I wound not be able to recall and recreate it in my grammatical thoughts; such a thought, despite it did occur, would be impossible to perceive at any later time.
Ad 2. If there is anything these stories relate to (other than other stories, via metaphor or other similarity, “isomorphism”, “embedding” etc.) these are what I so far loosely call observations, to avoid the term qualia. This however seems to lead to something similar (if not identical) to analytical hierarchy, and while some speculations could be made, this would contradict the arbitrarity of the “map” I might hold. Currently I see no way to have no ordering of maps and yet anything on the bottom. The only way to consider observations is as some relation between myself and the currently held map.
Ad 3. It seems to me the stories need to be “locally consistent”, but I am obviously blinded with mathematical education where excluded middle is treated as the holiest rule of all. But I only feel uncomfortable with plain inconsistencies, and seem to be absolutely unmoved by the incompatibilities of states of affairs which are either “distant enough” or incommensurable. Maybe learning a bit of paraconsistent logics will help to grasp that at some point?
Concerning completeness I am pretty sure the stories can never be complete, that it is the inherent property “of all my stories”. To put it in yet other words: for every story I find myself in, there are questions which have no answer, the image is always incomplete (metaphorically and literally). This might be a consequence of finiteness of my imagination, and expressive power of the (meta)language I use.
There are some nice consequences of this “perfectly antirealist” position, namely:
There are no ontological problems like mind-body dualism, identity problems (Theseus ship, breaking glasses etc.) — these are merely the boundaries of one story which I (or anyone else, if there is anyone else) desperately and hopelessly try to fit into another story. Which obviously causes insurmountable friction — since they are different stories!
There is no provable vs true problem or distinction. On certain days I have this impression I lost my understanding of “true but unprovable” (although in case of Godel’s sentences it is still okay as they are “a cheap trick” in a way). Perhaps I am turning into intuitionist and some kind of formalized ultrafinitist (but why not? it brings so much solace really). These positions are not only related to mathematical objects, but to all abstractions — and my point is there are no “non-abstract things”, therefore philosophies of mathematics do extend to pretty much everything.
There are no problems of “absolute truth”, perfectly in line with phenomena of undecidability, independence (forcing) etc. This feels really good and sheds some light on why Lakatos eventually failed in his program of re-realization of Kuhn’s revolutions.
Finally, the best part is (although that might very well be a mere coincidence) that this position just makes me very happy.